Linear Independence#
We start by returning the question: when does \(A\mathbf{x} = \mathbf{b}\) have a solution \(\mathbf{x}\)?
That is, when is \(A\mathbf{x} = \mathbf{b}\) consistent?
In the last lecture, we learned that \(A{\bf x} = {\bf b}\) is consistent if and only if \(\bf b\) lies in the span of the columns of \(A.\)
As an example, we saw for the following matrix \(A\):
\[\begin{split}A = \left[\begin{array}{rrr}1&3&4\\-4&2&-6\\-3&-2&-7\end{array}\right]\end{split}\]
\(A{\bf x} = {\bf b}\) is not consistent for all \({\bf b}\).
We realized that was because the span of \(A\)’s columns is not all of \(\mathbb{R}^3\), but rather only a part of \(\mathbb{R}^3\) – namely, a plane lying within \(\mathbb{R}^3\).
So, when \(\bf b\) does not lie in that plane, then \(A{\bf x} = {\bf b}\) is not consistent and has no solution.